Finite totally geodesic hypersurfaces in curved manifolds proven.
problem Characterizing totally geodesic hypersurfaces in curved manifolds.
method Analytic Riemannian manifold analysis with negative sectional curvature.
result Closed manifolds with negative curvature have only finitely many totally geodesic hypersurfaces.
Classifies totally geodesic submanifolds in specific geometric spaces.
problem Identifying totally geodesic submanifolds in homogeneous nearly Kähler 6-manifolds and their G2-cones.
method Developed new techniques for studying totally geodesic submanifolds in analytic Riemannian manifolds, homogeneous spaces, and Riemannian cones.
result Obtained a classification of totally geodesic submanifolds in homogeneous nearly Kähler 6-manifolds and their G2-cones.
Research confirms a conjecture about complex manifolds with total Betti number three.
problem Understanding the minimal total Betti number of closed almost complex manifolds.
method Analyzing properties of almost complex manifolds and using topological results.
result The only simply connected closed complex manifold with total Betti number three is the complex projective plane.
Totally geodesic surfaces found in knots and links.
problem Finding totally geodesic surfaces in knots and links.
method Constructing infinite families of knots and links with totally geodesic spanning surfaces in various 3-manifolds.
result Infinite families of knots and links with totally geodesic spanning surfaces in multiple 3-manifolds.
Study Clairaut anti-invariant submersions on nearly Kaehler manifolds.
problem Characterize geometric properties of Clairaut anti-invariant submersions.
method Investigate conditions for total geodesic maps and totally umbilical fibers.
result Conditions for Clairaut anti-invariant submersions to be totally geodesic.
We study non-degenerate, totally umbilical surfaces of a special class of pseudo-Riemannian manifolds, namely Walker three-manifolds. We show that such surfaces are either one of a totally geodesic family described by Calvaruso and Van der Veken or the ambient manifold must be locally conformally flat (here the surface…
Formula for comparing total curvature in curved spaces, with applications to isoperimetric problems.
problem Comparing total curvature in curved spaces and solving isoperimetric problems.
method Explicit formula for total curvature of level sets, applications to isoperimetric problems.
result Explicit formula for comparing total curvature in Riemannian manifolds.
The study finds totally geodesic surfaces in hyperbolic 3-manifolds and verifies a conjecture.
problem Finding totally geodesic surfaces in hyperbolic 3-manifolds.
method Developed algorithms to determine and verify the existence of totally geodesic surfaces.
result Discovered nine 3-manifolds with totally geodesic surfaces and verified Menasco-Reid's conjecture for knots up to 12 crossings.
Study of CR-submanifolds in various Lorentzian manifolds.
problem Exploring CR-submanifolds in different Lorentzian structures.
method Analyzing properties and results of CR-submanifolds in LCS, LP-cosymplectic, S, and GKM manifolds.
result Obtained results on totally umbilical and geodesic CR-submanifolds.
Arithmetic 3-manifolds with infinite geodesics are proven.
problem Arithmeticity of 3-manifolds with infinitely many geodesics.
method Analysis of totally geodesic surfaces in hyperbolic 3-manifolds.
result Arithmetic 3-manifolds with infinite geodesics are proven.
The purpose of this paper is to classify totally umbilical slant submanifolds of a Kenmotsu manifold. We prove that a totally umbilical slant submanifold M of a Kenmotsu manifold Mˉ is either invariant or anti-invariant or dimM=1 or the mean curvature vector H of M lies in the invariant normal subbundle.…
Totally geodesic dual leaves on curved manifolds are also curved.
problem Characterizing dual leaves of nonnegatively curved polar manifolds.
method Proving dual leaves are totally geodesic and closed, and inducing a Riemannian submersion.
result Dual leaves of nonnegatively curved polar manifolds are themselves nonnegatively curved and totally geodesic.
Study on CR-lightlike submanifolds in golden semi-Riemannian manifolds.
problem Exploring properties of CR-lightlike submanifolds in golden semi-Riemannian manifolds.
method Definition and investigation of properties of geodesic CR-submanifolds.
result Obtained results for totally geodesic and totally umbilical CR-submanifolds.
We introduce semi-invariant Riemannian submersions from almost Hermitian manifolds onto Riemannian manifolds. We give examples, investigate the geometry of foliations which are arisen from the definition of a Riemannian submersion and find necessary-sufficient conditions for total manifold to be locally product Riemann…
The present paper deals with the study of totally real submanifolds and C-totally real submanifolds of (LCS)n-manifolds with respect to Levi-Civita connection as well as quarter symmetric metric connection. It is proved that scalar curvature of C-totally real submanifolds of (LCS)n-manifold …
Totally umbilical hypersurfaces in Spin^c manifolds with special spinors have constant mean curvature.
problem Characterizing totally umbilical hypersurfaces in Spin^c manifolds with specific spinor fields.
method Proving constant mean curvature for hypersurfaces carrying parallel, real or imaginary Killing spinors.
result Results extend to Spin^c case, generalizing O. Kowalski's theorem.
Classifies totally geodesic submanifolds and polar actions on Stiefel manifolds.
problem Classifying totally geodesic submanifolds and polar actions on Stiefel manifolds.
method Classification through polar actions and cohomogeneity-one actions.
result Classification of orbits of polar actions on Stiefel manifolds.
Totally umbilic surfaces in hyperbolic 3-manifolds are constructed and characterized.
problem Characterizing totally umbilic surfaces in hyperbolic 3-manifolds of finite volume.
method Construction and characterization of surfaces based on their properties and embedding conditions.
result Characterization of totally umbilic surfaces in hyperbolic 3-manifolds of finite volume.
Study shows certain 4D hyperbolic links don't contain geodesic 3-manifolds.
problem Proving certain hyperbolic link complements don't contain geodesic 3-manifolds.
method Analyzing hyperbolic link complements of 2-tori in S^4.
result Proves certain hyperbolic link complements do not contain closed embedded totally geodesic hyperbolic 3-manifolds.
The study examines spacelike foliations on Lorentz manifolds under specific conditions.
problem Investigating geometric properties of spacelike foliations on Lorentz manifolds.
method Analyzing conditions for stability, total geodesy, and total umbilicity of foliation leaves.
result Conditions for the stability, total geodesy, and total umbilicity of spacelike foliation leaves are established.
Study on Clairaut maps from nearly Kahler to Riemannian manifolds.
problem Characterizing Clairaut maps from nearly Kahler manifolds.
method Analyzing conditions for Clairaut maps to be totally geodesic foliations.
result Non-trivial examples of Clairaut maps are provided.
Study on nonspin manifolds with spin boundary, showing nonconnectedness and nontrivial fundamental group.
problem Understanding spaces of positive scalar curvature metrics on totally nonspin manifolds.
method Analysis of positive scalar curvature metrics on manifolds with spin boundary, using propagation techniques.
result Spaces of positive scalar curvature metrics are not connected and have nontrivial fundamental groups for certain dimensions.
Totally geodesic null hypersurfaces found in Lorentzian manifolds.
problem Characterizing null hypersurfaces in Lorentzian manifolds.
method Analyzing light-like geodesically complete Lorentzian manifolds with null energy condition.
result Null hypersurfaces are totally geodesic.
The paper characterizes submersions with Ricci soliton total manifolds and their properties.
problem Characterizing submersions with Ricci soliton total manifolds.
method Analyzing Riemannian submersions and their properties, focusing on fiber and target manifolds.
result Necessary conditions for the target manifold to be a Ricci soliton and characterizations of harmonicity.
Study geometric properties of spacelike foliations on Lorentz manifolds.
problem Investigate conditions for spacelike foliations to be totally umbilical or geodesic.
method Develop an equation relating foliation to ambient manifold, apply Maximum principle.
result Obtain an obstruction for totally geodesic foliations in spacetimes with positive Ricci curvature.
We prove an inequality between the sum of the Betti numbers of a complex projective manifold and its total curvature, and we characterize the complex projective manifolds whose total curvature is minimal. These results extend the classical theorems of Chern and Lashof to complex projective space.
This notes explores angle structures on ideally triangulated compact 3-manifolds with high genus boundary. We show that the existence of angle structures implies the existence of a hyperbolic metric with totally geodesic boundary, and conversely each hyperbolic 3-manifold with totally geodesic boundary has an ideal…
We prove that the total CR Q-curvature vanishes for any compact strictly pseudoconvex CR manifold. We also prove the formal self-adjointness of the P′-operator and the CR invariance of the total Q′-curvature for any pseudo-Einstein manifold without the assumption that it bounds a Stein manifold.
We give a full description of totally geodesic submanifolds in the tangent bundle of a Riemannian 2-manifold of constant curvature and present a new class of a cylinder-type totally geodesic submanifolds in the general case.
We obtain an exhaustive classification of totally umbilical surfaces in unimodular and non-unimodular simply-connected 3-dimensional Lie groups endowed with arbitrary left-invariant Riemannian metrics. This completes the classification of totally umbilical surfaces in homogeneous Riemannian 3-manifolds.
On asymptotically flat and asymptotically hyperbolic manifolds, by evaluating the total mass via the Ricci tensor, we show that the limits of certain Brown-York type and Hawking type quasi-local mass integrals equal the total mass of the manifold in all dimensions.
Study Dirichlet-to-Neumann maps on manifolds, focusing on covering and total spaces.
problem Understanding the Steklov spectrum of covering and total spaces.
method Analyzing Dirichlet-to-Neumann maps on Riemannian manifolds with boundary and bounded geometry.
result Existence and properties of the bottom of the Dirichlet spectrum on covering and total spaces.
Defines and proves CR invariants on five-manifolds.
problem Defines and studies CR invariants on CR five-manifolds.
method Defines global secondary CR invariants and proves their linear combination.
result Any global secondary CR invariant is a linear combination of total Q′-curvature, total I′-curvature, and a local CR invariant. We classify totally geodesic and parallel hypersurfaces of four-dimensional non-reductive homogeneous pseudo-Riemannian manifolds.
We prove that a Riemannian product of type M x R (where R denotes the Euclidean line) admits totally umbilical hypersurfaces if and only if M has locally the structure of a warped product and we give a complete description of the totally umbilical hypersurfaces in this case. Moreover, we give a necessary and sufficient…
Study on null hypersurfaces in complex contact manifolds.
problem Characterizing null hypersurfaces in indefinite complex contact manifolds.
method Proved classification results for various null hypersurfaces and characterized the ambient space.
result The ambient complex contact manifold must have constant GH-sectional curvature of −3 for certain null hypersurfaces. Totally geodesic sections found in polar actions.
problem Understanding sections of polar actions on Riemannian manifolds.
method Elementary proof of a folklore result.
result Sections of polar actions are totally geodesic.
We derive variational formulas for the total Q-prime curvature under the deformation of strictly pseudoconvex domains in a complex manifold. We also show that the total Q-prime curvature agrees with the renormalized volume of such domains with respect to the complete Einstein-Kähler metric. In the appendix, by Rod Gove…
The authors showed in a preceding paper that in a connected locally harmonic manifold, the volume of a tube of small radius about a regularly parameterized simple arc depends only on the length of the arc and the radius. In this paper, we show that this property characterizes harmonic manifolds even if it is assumed on…
Bi-slant submersion generalizes slant and semi-slant submersion.
problem Generalizing submersion types in Riemannian geometry.
method Introducing and studying bi-slant ξ⊥-Riemannian submersions. result Necessary and sufficient conditions for totally geodesic submersion.
New formulas compare total mean curvatures of nested hypersurfaces.
problem Computing total mean curvatures of nested hypersurfaces.
method Developed differential forms based on Chern's work to compare curvatures.
result Quicker proof of recent result on total mean curvatures.
Closed surfaces minimize total curvature in curved spaces.
problem Minimizing total curvature in curved spaces.
method Isometric embedding via holonomy and Pogorelov's theory.
result Closed surfaces bound flat convex bodies.
Researchers decompose hyperbolic n-manifolds with totally geodesic boundaries into polyhedral cells.
problem Decomposing hyperbolic n-manifolds with totally geodesic boundaries into polyhedral cells.
method Two different approaches to demonstrate the existence of polyhedral decompositions.
result The number of polyhedral decompositions of M is finite. Sharp curvature inequality extends Euclidean isoperimetric inequality to Cartan-Hadamard manifolds.
problem Proving the Euclidean isoperimetric inequality on Cartan-Hadamard manifolds with nullity.
method Using the Chern-Gauss-Bonnet theorem to establish a sharp inequality for total curvature.
result The Euclidean isoperimetric inequality extends to Cartan-Hadamard manifolds with nullity.
In this paper we investigate what kind of manifolds arise as the total spaces of iterated S1-bundles. A real Bott tower studied in \cite{CMO}, \cite{KM} and \cite{KN} is an example of an iterated S1-bundle. We show that the total space of an iterated S1-bundle is homeomorphic to an infra-nilmanifold. A real Bo…
A bound on surface area in Riemannian manifolds without totally geodesic surfaces.
problem Bounding the area of surfaces in Riemannian manifolds without totally geodesic surfaces.
method Using extrinsic curvature energy to bound surface area.
result The area of any complete surface immersed into M is bounded by a multiple of its extrinsic curvature energy. Study on random surfaces in hyperbolic 3-manifolds, focusing on geometric and topological properties.
problem Distribution of nearly geodesic surfaces in hyperbolic 3-manifolds.
method Invariant measures on the Grassmann bundle G(M) derived from limits of random minimal surfaces.
result Topological limiting measures are totally scarring if M contains a totally geodesic subsurface, while geometrical limiting measures are not.
Complex hyperbolic manifolds with many totally geodesic submanifolds are arithmetic.
problem Characterizing arithmeticity of complex hyperbolic manifolds with certain submanifolds.
method Developing superrigidity theorems for complex hyperbolic lattices and proving nonexistence of certain maps.
result Finite volume complex hyperbolic n-manifolds containing infinitely many maximal totally geodesic submanifolds of dimension at least two are arithmetic.